Leo Lam © Signals and Systems EE235

Leo Lam © Stanford The Stanford Linear Accelerator Center was known as SLAC, until the big earthquake, when it became known as SPLAC. SPLAC? Stanford Piecewise Linear Accelerator.

Leo Lam © Today’s menu Today: Linear, Constant-Coefficient Differential Equation – Particular Solution

Zero-state output of LTI system Leo Lam © Response to our input x(t) LTI system: characterize the zero-state with h(t) Initial conditions are zero (characterizing zero-state output) Zero-state output: Total response(t)=Zero-input response (t)+Zero-state output(t) T  (t) h(t)

Zero-state output of LTI system Leo Lam © Zero-input response: Zero-state output: Total response: Total response(t)=Zero-input response (t)+Zero-state output(t) “Zero-state”:  (t) is an input only at t=0 Also called: Particular Solution (PS) or Forced Solution

Zero-state output of LTI system Leo Lam © Finding zero-state output (Particular Solution) Solve: Or: Guess and check Guess based on x(t)

Trial solutions for Particular Solutions Leo Lam © Guess based on x(t) Input signal for time t> 0 x(t) Guess for the particular function y P

Particular Solution (example) Leo Lam © Find the PS (All initial conditions = 0): Looking at the table: Guess: Its derivatives:

Particular Solution (example) Leo Lam © Substitute with its derivatives: Compare:

Particular Solution (example) Leo Lam © From We get: And so:

Particular Solution (example) Leo Lam © Note this PS does not satisfy the initial conditions! Not 0!

Natural Response (doing it backwards) Leo Lam © Guess: Characteristic equation: Therefore:

Complete solution (example) Leo Lam © We have Complete Sol n : Derivative:

Complete solution (example) Leo Lam © Last step: Find C 1 and C 2 Complete Sol n : Derivative: Subtituting: Two equations Two unknowns

Complete solution (example) Leo Lam © Last step: Find C 1 and C 2 Solving: Subtitute back: Then add u(t): y n ( t ) y p ( t ) y ( t )

Another example Leo Lam © Solve: Homogeneous equation for natural response: Characteristic Equation: Therefore: Input x(t)

Another example Leo Lam © Solve: Particular Solution for Table lookup: Subtituting: Solving: b=-1, =-2 No change in frequency! Input signal for time t> 0 x(t) Guess for the particular function y P

Another example Leo Lam © Solve: Total response: Solve for C with given initial condition y(0)=3 Tada!

Stability for LCCDE Leo Lam © Stable with all Re( j <0 Given: A negative means decaying exponentials Characteristic modes

Stability for LCCDE Leo Lam © Graphically Stable with all Re( j )<0 “Marginally Stable” if Re( j )=0 IAOW: BIBO Stable iff Re(eigenvalues)<0 Im Re Roots over here are stable

Leo Lam © Summary Differential equation as LTI system

Leo Lam © Next topic! Fourier Series –1 st topic “Orthogonality”

Fourier Series: Introduction Leo Lam © Fourier Series/Transform: Build signals out of complex exponentials –Periodic signals –Extend to more general signals Why? –Convolution: hard –Multiplication: easy (frequency domain) Some signals are more easily handled in frequency domain

Fourier Series: Why Complex Exp? Leo Lam © Complex exponentials are nice signals –Eigenfunctions to LTI –Convolution (in t)  Multiplication (in ) Frequency: directly related to sensory Harmonics: Orthogonality (later today) –Orthogonality simplifies math

The beauty of Fourier Series Leo Lam © Recall: Write x(t) in terms of e st (Fourier/Laplace Transform) The input is a sum of weighted shifted impulses The output is a sum of weighted shifted impulses S Special input:

The beauty of Fourier Series Leo Lam © Write x(t) in terms of e st (Fourier/Laplace Transform) Make life easier by approximation: Output: LTI Sum of weighted eigenfunctions Sum of scaled weighted eigenfunctions

Definition: Approximation error Leo Lam © Approximating f(t) by cx(t): Choose c so f(t) is as close to cx(t) as possible Minimizing the error energy: Which gives: error Dot-product

Dot product: review Leo Lam © Dot product between two vectors Vectors (and signals) are orthogonal if their dot product is zero. f x Angle between the two vectors

Vector orthogonality Leo Lam © Vectors (and signals) are orthogonal if their dot product is zero. Dot product: length of x projected onto a unit vector f Orthogonal: cos()=0 Perpendicular vectors=no projection f x f x Key idea

Visualize dot product Leo Lam © Let a x be the x component of a Let a y be the y component of a Take dot product of a and b In general, for d-dimensional a and b x-axis a y-axis b

Visualize dot product Leo Lam © In general, for d-dimensional a and b For signals f(t) and x(t) For signals f(t) and x(t) to be orthogonal from t 1 to t 2 For complex signals Fancy word: What does it mean physically?